The proof corresponding to the general case is complete. Assume in additional that the operator \(R_{\tilde{W}}\)  is normal. Let us show that the operator \(V:=(R_{\tilde{W}}+R^{\ast }_{\tilde{W}})/2\)  has a complete orthonormal system of the eigenvectors. Using formula (53) [1]}, we get
\(V^{\!^{-1}}=2H^ \frac{1 }{2} (I+B^{2}) H^{ \frac{1}{2}}.\) 
